Question

The difference between roots of the quadratic equation x^2+x+c=0 is 6. find c.

Answer 1

Answer:

$\displaystyle c = -\frac{35}{4} = -8.75$.

Step-by-step explanation:

Let the smaller root to this equation be $m$. The larger one will equal $m + 6$.

By the factor theorem, this equation is equivalent to

$a(x - m)(x - (m+6))= 0$, where $a \ne 0$.

Expand this expression:

$a\cdot x^{2} - a(2m + 6)\cdot x + a(m^{2} + 6m) =0$.

This equation and the one in the question shall differ only by the multiple of a non-zero constant. It will be helpful if that constant is equal to $1$. That way, all constants in the two equations will be equal; $(m^{2} + 6m)$ will  be equal to $c$.

Compare this equation and the one in the question:

The coefficient of $x^{2}$ in the question is $1$ (which is omitted.) The coefficient of $x^{2}$ in this equation is $a$. If all corresponding coefficients in the two equations are equal to each other, these two coefficients shall also be equal to each other. Therefore $a = 1$.

This equation will become:

$x^{2} - (2m + 6)\cdot x + (m^{2} + 6m) =0$.

Similarly, for the coefficient of $x$,

$\displaystyle -(2m +6) = 1$.

$\displaystyle m = -\frac{7}{2}$.

This equation will become:

$x^{2} + x + \underbrace{\left(-\frac{35}{4}\right)}_{c} =0$.

$c$ is the value of the constant term of this quadratic equation.